Denote by $k$ an algebraically closed field. Can one produce a DVR $A$ over $k$ such that

- the fraction field of $A$ has an automomorphism not preserving $A$
- no non-trivial field extension of $k$ maps, as a $k$-algebra, to $A$?

This question is in part inspired by this post (though I guess the connection is not entirely clear, I will try to clarify if this gets responses).

If $k((x))$ has an automorphism not preserving $k[[x]]$, that would mean a positive answer to this question. I do not know if such automorphism exists.